What are the topological properties of the Cantor set if a point is added at the center of every empty segment?
Suppose that when the segment $\left(\frac13,\frac23\right)$ is deleted from $[0,1]$, a point is also not deleted at its centre $\frac12$ and the process then proceeds by repetition (i.e. deleting all but a central point at each step) as in the normal construction.
What are the topological properties of this set?
It's easy to see that at least an infinite, binary rooted tree of points is added. But what about the limit points? Is it still totally disconnected and measure zero?
The result is that you've added countably many isolated points to the Cantor set. In particular, this set is still totally disconnected and measure zero.